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Thermal Flux in Unsteady Rayleigh-B\'{E} Nard

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Thermal Flux in Unsteady Rayleigh-B\'{E} Nard Thermal flux in unsteady Rayleigh-Benard´ magnetoconvection a a, Sandip Das , Krishna Kumar ∗ a Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India Abstract We present results of numerical investigation on thermal flux in Rayleigh-B´enard magnetoconvection in the presence of a uniform vertical magnetic field. We have studied thermal flux in different viscous fluids with a range of Prandtl number (0.1 Pr < 6.5) and a range of Chandrasekhar number (50 Q 2.5 104). The power spectral density of the Nusselt number≤ varies with 2 ≤ ≤ × frequency f approximately as f − . The probability distribution function of the fluctuating part of the Nusselt number is nearly normal distribution with slight asymmetric tails. For a fixed value the Rayleigh number Ra, the time averaged Nusselt number Nu(Q) decreases logarithmically with Chandrasekhar number for Q > Qc, which depends on Ra and Pr. The reduced Nusselt h i number Nur = Nu(Q) / Nu(0) rises sharply, reaches a maximum slightly above unity and then start decreasing very slowly to unity as the valueh of ai dimensionlessh i parameter √Ra/(Q Pr) is raised. The probability distribution function of the local thermal flux in the vertical direction is found to be asymmetric and non-Gaussian with a cusp at its maximum. Keywords: Magnetoconvection, Rayleigh number, Chandrasekhar number, Nusselt number, local heat flux, thermal boundary layer, Power spectral density, nanofluids 1. Introduction Experimentson the measurementof thermal flux in magneto- convection in metallic fluids [21–23] showed that the transport The understanding of heat flux in magnetoconvective flows of heat across the fluid layer in turbulent magnetoconvection is a topic of intense research due to its potential industrial ap- was reduced significantly and the fluid flow was affected [20]. plications in nanofluids [1–3], biofluids [4], electro-chemical It was also found that the time averaged Nusselt number Nu process [5] and material processing research [5–7] in addition showed scaling behaviour with Rayleigh number Ra [21,h22].i to its relevance in traditional areas like geophysics [8–12] and The scaling exponent was found to depend on the strength of astrophysics [13–16]. A thermally stratified system, where a the applied magnetic field. A uniform nanofluid consists of ho- thin horizontal layer of a fluid is subjected to an adverse tem- mogeneous suspension of metallic nanoparticles in an ordinary perature gradient and simultaneously subjected to a uniform fluid, which is also known as a base/carrier fluid. The viscous, magnetic field, is known as Rayleigh-B´enard magnetoconvec- thermal, electrical and magnetic properties of a nanofluid de- tion (RBM) [17–20]. Chandrasekhar [17] analysed the linear pend on the properties of the base fluid as well as the properties problem of thermal convection in a homogeneous fluid. He of suspended nanoparticles. There is hardly any work on the showed that a uniform vertical magnetic field delays the con- role of magnetic field on the heat flux in unsteady flows in flu- vective flow. In addition, he showed that the onset of convection ids including nanofluids, liquid crystals and metallic fluids. is always stationary if Pr is greater than Pm. In this article we present results of numerical simulations on The dynamics of RBM is governed by four dimensionless both global and local heat fluxes in RBM with a uniform verti- quantities: cal magnetic field in water based nanofluids with low dilution (1) Rayleigh number Ra, which is the relative measure of the of non-magnetic spherical nanoparticles. We have computed buoyancy force over the dissipative force, Nusselt number Nu(Q), which is a measure of the global heat (2) Chandrasekhar’s number Q, which is a measure of the flux for non-zero value of Chandrasekhar’s number Q, which arXiv:1909.11908v2 [physics.flu-dyn] 26 Aug 2020 strength of the Lorentz force, is a ratio of the Lorentz force per unit volume to the drag force (3) the thermal Prandtl number Pr = ν/κ is a ratio of the effec- due to magneto-viscous effect. The time averaged Nusselt num- tive kinematic viscosity ν and the effective thermal diffusivity ber Nu(Q) increases slowly with Chandrasekhar number Q κ, and for smallerh i values of Q. The effective Prandtl number of fluid (4) the effective magnetic Prandtl number Pm = σµ ν, where 0 is varied from 0.1 Pr 6.4. As soon as Q is raised above σ is the electrical conductivity of the fluid and µ is the mag- 0 a critical value Q ,≤ which≤ depends on Ra and Pr, Nu(Q) de- netic permeability of air. The magnetic diffusivity of the fluid c creases logarithmically with Q for a fixed value of Ra.h Wei have is defined as η = 1/(µ0σ). also plotted the variation of the reduced Nusselt number Nur = Nu(Q) / Nu(0) with a dimensionless parameter √Ra/(QPr), whichh isi ah ratioi of the buoyancy and Lorentz forces. For flu- ∗Corresponding author Email address: [email protected] ( Krishna Kumar) ids with Prandtl number Pr 4.0, Nu increases sharply with ≤ r Preprint submitted to International Journal of Heat and Mass Transfer August 28, 2020 the dimensionless parameter √Ra/(QPr). It attains a maximum Similarly the effective thermal diffusion coefficient κ = K/(ρcV ) slightly above unity and then begins decreasing slowly towards of a nanofluid with spherical non-magnetic metallic particles unity, as √Ra/(QPr) is further raised. The probability distri- may be computed using the expression: bution of fluctuations in the Nusselt number is close to nor- K f mal with slightly asymmetric tails. The probability distribution κ = [(1 φ)(ρcV) f + φ(ρcV )p] functions (PDF) of the local heat fluxes in the vertical direc- − (Kp + 2K f ) 2φ(K f Kp) tion are found to be non-Gaussian with a cusp at their maxima. − − . (8) PDFs are asymmetric about their maxima and have exponential × " (Kp + 2K f ) + φ(K f Kp) # − tails. Initially the fluid is at rest and the heat flux across the fluid layer is only due to conduction. The lower boundary of the nanofluid is maintained at temperature T , while the upper boundary is 2. Hydromagnetic System b maintained at temperature Tu = Tb ∆T. Here, β = (Tu − − Tb)/d = ∆T /d < 0. The steady state temperature profile Ts(z), We consider a thin horizontal layer of a homogeneous nano- density stratification ρs(z) and the pressure field Ps(z) across the fluid of effective density ρ and thickness d, effective thermal nanofluid in conduction state [17] are given by, expansion coefficient α and effective electrical conductivity σ and subjected to an adverse temperature gradient β in the pres- Ts(z) = Tb + βz, (9) = ence of a uniform magnetic field B0 B0e3 directed along the ρs(z) = ρ0 [1 + α (Tb Ts(z))] , (10) vertical direction. Here e3 is a unit vector in the vertically up- − 1 2 ward direction. The effective density ρ and the electrical con- Ps(z) = P0 ρ0g z + αβz , (11) − 2 ! ductivity σ are expressed [3] as: where Tb and ρ0 are the reference values of the temperature and ρ = (1 φ)ρ f + φρp, (1) density fields at the bottom surface of the nanofluid. P is a − 0 σ = (1 φ)σ f + φσp, (2) constant, which includes the magnetic pressure. As soon as β − is raised above a critical value βc, the basic state of conduction where φ is the volume fraction of the suspended spherically becomes unstable and convective flow (v , 0) begins. All the shaped nanoparticles of density ρp and electrical conductivity fields are perturbed and may be written as: σp in a base fluid of density ρ f and electrical conductivity σ f . ρs(z) ρ˜(x, y, z, t) = ρs(z) + δρ(x, y, z, t), (12) We may express the products ρα and ρcV for nanofluids [3] as: → Ts(z) T(x, y, z, t) = Ts(z) + θ(x, y, z, t), (13) (ρα) = (1 φ)(ρα) + φ(ρα) , (3) → f p Ps(z) P(x, y, z, t) = Ps(z) + p(x, y, z, t), (14) − → (ρcV ) = (1 φ)(ρcV) f + φ(ρcV)p, (4) B B(x, y, z, t) = B + b(x, y, z, t). (15) − 0 → 0 where cV stands for the effective specific heat of nanofluid at All length scales are measuredin units of the fluid thickness d constant volume. The effective thermal conductivity K of a and time is measuredin unitsof the free-fall time τ f = 1/ √αgβ, nanofluid [24] with spherical nanoparticles of thermal conduc- where g is the acceleration due to gravity. The fluid veloc- T tivity Kp in a base fluid of thermal conductivity K f is expressed ity v(x, y, z, t) = (v1, v2, v3) , the perturbation in pressure due as to flow p(x, y, x, t), the convective temperature θ(x, y, z, t) and the induced magnetic field b(x, y, z, t) are made dimensionless (Kp + 2K f ) 2φ(K f Kp) 2 2 K = K f − − . (5) by αgβd , ρ0αgβd , βd and νσµ0 B0, respectively. The value " (Kp + 2K f ) + φ(K f Kp) # − of thep effective magnetic Prandtl number Pm is of the order of 10 5 or less for terrestrial fluids including nanofluids. We Following Brinkman [25], the effective dynamic viscosity µ of − therefore set the value of Pm equal to zero in this work. This a nanofluid may be modelled as: makes the induced magnetic field b a slaved variable. The RBM 2.5 in nanofluids is then described by the following dimensionless µ = µ f (1 φ)− , (6) − equations: where µ f is the dynamic viscosity of the base fluid.
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